On cohomology of the Lie superalgebra D(2, 1 ; \alpha)

We describe the infinitesimal deformations of the standard embedding of the Lie superalgebra $D(2,1;\alpha )$ into the Poisson superalgebra of pseudodifferential symbols on $S^{1\mathrm{\mid }2}$. We show that for the standard embedding of $D(2,1;\alpha )$ into the Poisson superalgebra of differential operators on $S^{1\mathrm{\mid }2}$, the infinitesimal deformations correspond to formal deformations. For the embedding of $D(2,1;\alpha )$ into the derived contact superconformal algebra $K^{\mathrm{\prime }}(4)$, the infinitesimal deformations are formal deformations.